A299409 Solution (e(n)) of the system of 5 complementary equations in Comments.

10, 26, 45, 62, 78, 94, 114, 130, 146, 162, 180, 198, 214, 230, 248, 266, 282, 298, 317, 334, 350, 366, 386, 402, 418, 434, 451, 470, 486, 502, 520, 538, 554, 570, 589, 606, 622, 638, 658, 674, 690, 706, 725, 742, 758, 774, 792, 810, 826, 842, 861, 878, 894

Offset

0,1

Comments

Define sequences a(n), b(n), c(n), d(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 3:

a(n) = least new;

b(n) = least new;

c(n) = least new;

d(n) = least new;

e(n) = a(n) + b(n) + c(n) + d(n);

where "least new k" means the least positive integer not yet placed.

***

Conjecture: for all n >= 0,

0 <= 17n - 11 - 4 a(n) <= 4

0 <= 17n - 7 - 4 b(n) <= 4

0 <= 17n - 3 - 4 c(n) <= 3

0 <= 17n + 1 - 4 d(n) <= 3

0 <= 17n - 5 - e(n) <= 3

***

The sequences a,b,c,d,e partition the positive integers. The sequence e can be called the "anti-tetranacci sequence"; see A075326 (anti-Fibonacci numbers) and A265389 (anti-tribonacci numbers).

Links

Clark Kimberling, Table of n, a(n) for n = 0..1000

Example

n:   0  1   2    3   4   5   6   7   8   9
a:   1  5   9   14  18  22  27  31  35  39
b:   2  6   11  15  19  23  28  32  36  40
c:   3  7   12  16  20  24  29  33  37  41
d:   4  8   13  17  21  25  30  34  38  42
e:  10  26  45  62  78  94 114 130 146 162

Mathematica

(* Program 1: sequences a, b, c, d, e generated from the complementary equations *)
z = 200;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {3}; d = {4}; e = {}; AppendTo[e,
Last[a] + Last[b] + Last[c] + Last[d]];
Do[{AppendTo[a, mex[Flatten[{a, b, c, d, e}], 1]],
   AppendTo[b, mex[Flatten[{a, b, c, d, e}], 1]],
   AppendTo[c, mex[Flatten[{a, b, c, d, e}], 1]],
   AppendTo[d, mex[Flatten[{a, b, c, d, e}], 1]],
   AppendTo[e, Last[a] + Last[b] + Last[c] + Last[d]]}, {z}];
Take[a, 100]  (* A299405 *)
Take[b, 100]  (* A299637 *)
Take[c, 100]  (* A299638 *)
Take[d, 100]  (* A299641 *)
Take[e, 100]  (* A299409 *)
(* Program 2: sequence e generated by iterating a morphism *)
morph = Nest[Flatten[# /. Thread[{0, 1, 2, 3} -> {{2, 3, 3, 1}, {2, 3, 2, 1}, {2, 3, 1, 1}, {2, 3, 0, 1}}]] &, {0}, 9];
A299409 = Accumulate[Prepend[Drop[Flatten[morph /. Thread[{0, 1, 2, 3} -> {{1, 1, 2, 4}, {1, 1, 3, 3}, {1, 1, 4, 2}, {1, 1, 5, 1}}]], 1] + 15, 10]];
Take[A299409, 100]  (* Peter J. C. Moses, May 04 2018 *)

Crossrefs

Cf. A036554, A299634, A299405, A299637, A299638, A299641.

Sequence in context: A223451 A229309 A044452 * A198017 A137351 A134406

Adjacent sequences: A299406 A299407 A299408 * A299410 A299411 A299412

Keyword

nonn,easy

Author

Clark Kimberling, Apr 22 2018

Status

approved